Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: A == (~P)*D*P
True
The matrix P may contain zero rows corresponding to eigenvalues for which the algebraic multiplicity is
greater than the geometric multiplicity. In these cases, the matrix is not diagonalizable.
sage: A = jordan_block(2,3); A
[2 1 0]
[0 2 1]
[0 0 2]
sage: A = jordan_block(2,3)
sage: D, P = A.eigenmatrix_left()
sage: D
[2 0 0]
[0 2 0]
[0 0 2]
sage: P
[0 0 1]
[0 0 0]
[0 0 0]
sage: P*A == D*P
True
TESTS:
For matrices with floating point entries, some platforms will return eigenvectors that are negatives of
those returned by the majority of platforms. This test accounts for that possibility. Running this test
independently, without adjusting the eigenvectors could indicate this situation on your hardware.
sage: A = matrix(QQ, 3, 3, range(9))
sage: em = A.change_ring(RDF).eigenmatrix_left()
sage: evalues = em[0]; evalues.dense_matrix().zero_at(2e-15)
[ 13.3484692283 0.0 0.0]
[ 0.0 -1.34846922835 0.0]
[ 0.0 0.0 0.0]
sage: evectors = em[1];
sage: for i in range(3):
... scale = evectors[i,0].sign()
... evectors.rescale_row(i, scale)
sage: evectors
[ 0.440242867... 0.567868371... 0.695493875...]
[ 0.897878732... 0.278434036... -0.341010658...]
[ 0.408248290... -0.816496580... 0.408248290...]
eigenmatrix_right()
Return matrices D and P, where D is a diagonal matrix of eigenvalues and P is the corresponding matrix
where the columns are corresponding eigenvectors (or zero vectors) so that self*P = P*D.
EXAMPLES:
sage: A = matrix(QQ,3,3,range(9)); A
[0 1 2]
[3 4 5]
[6 7 8]
sage: D, P = A.eigenmatrix_right()
sage: D
[ 0 0 0]
[ 0 -1.348469228349535 0]
[ 0 0 13.34846922834954]
sage: P
156 Chapter 7. Base class for matrices, part 2